Chapter 1 Equations and Inequalities

Transcription

1 Chapter Equations and Inequalities Section. Check Point Exercises... The meaning of a [,,] by [,,] viewing rectangle is as follows: distance between x-axis minimum maximum tick x-value x-value marks [,, ] by distance between y-axis minimum maximum tick y-value y-value marks [,, ], y 7, y 6, y, y, y, y, y., y, y, y, y, y, y, y. a. The graph crosses the x-axis at (, ). Thus, the x-intercept is. The graph crosses the y-axis at (, ). Thus, the y-intercept is. b. The graph does not cross the x-axis. Thus, there is no x-intercept. The graph crosses the y-axis at (, ). Thus, the y-intercept is. c. The graph crosses the x- and y-axes at the origin (, ). Thus, the x-intercept is and the y-intercept is. 6. a. d n+ d () + 6 6% of marriages end in divorce after years when the wife is under 8 at the time of marriage. b. According to the line graph, 6% of marriages end in divorce after years when the wife is under 8 at the time of marriage. c. The mathematical model overestimates the actual percentage shown in the graph by %. Copyright Pearson Education, Inc.

2 Chapter Equations and Inequalities Concept and Vocabulary Check.. x-axis 9.. y-axis. origin. quadrants; four. x-coordinate; y-coordinate. 6. solution; satisfies 7. x-intercept ; zero 8. y-intercept; zero Exercise Set...., y 7, y, y, y, y, y, y , y, y, y, y, y, y, y Copyright Pearson Education, Inc.

3 Section. Graphs and Graphing Utilities 7.., y, y, y, y, y, y, y 7, y 6,y, y, y, y, y, y 6 9.., y, y, y, y, y, y, y, y, y, y, y, y, y, y Copyright Pearson Education, Inc.

4 Chapter Equations and Inequalities. 9. The graphs of Y and Y intersect at the points (,) and (,).. a. ; The graph intersects the x-axis at (, ). b. ; The graph intersects the y-axis at (, )., y,y, y 8, y 9, y 8, y, y. a., ; The graph intersects the x-axis at (, ) and (, ). b. ; The graph intersects the y-axis at (, ).. a. ; The graph intersects the x-axis at (, ). 7. b. none; The graph does not intersect the y-axis , y 7, y 8, y, y, y, y 8, y 7 9. (c) x-axis tick marks,,,,,,,,,, ; y-axis tick marks are the same.. (b); x-axis tick marks,,,,,,,, 6, 7, 8; y-axis tick marks,,,,,,,,, 6, 7. The equation that corresponds to Y in the table is (c), y x. We can tell because all of the points (,), (,), (,), (, ), (,), (,), and (, ) are on the line y x, but all are not on any of the others.. x ( x, y) (,) (,) (,) (,) (,) (,) (,). No. It passes through the point (, ). 7. (,) Copyright Pearson Education, Inc.

5 Section. Graphs and Graphing Utilities. x ( x, y),, ( ),,,,, ( ), 7. At age 8, women have the least number of awakenings, averaging about awakening per night. 9. The difference between the number of awakenings for -year-old men and women is about Answers will vary. 67. makes sense 69. does not make sense; Explanations will vary. Sample explanation: These three points are not collinear. 7. false; Changes to make the statement true will vary. A sample change is: The product of the coordinates of a point in quadrant III is also positive. 7. true 7. I, III 77. IV 79. (a) 8. (b) 8. (b) 8. (c). a. According to the line graph, % of seniors used marijuana in. b. is years after 98. M.n+ 7 M.() According to formula, 9.% of seniors used marijuana in. This underestimates the value in the graph by.%. c. According to the line graph, about 7% of seniors used alcohol in. d. is years after 98. A.9n+ 69 A.9() According to formula, 6.% of seniors used alcohol in. It is less than the estimate, although answers may vary. e. The minimum for marijuana was reached in 99. According to the line graph, about % of seniors used marijuana in ( x ) 7 ( x+ ) (6 ) 7 (6 + ) () 7 (8) 6 7, true x+ x x+ x ( x+ ) ( x ) + 6 x+ x + ( x ) + 9 ( x ) x x + ( x )(9) + 9x 7 9x Copyright Pearson Education, Inc.

6 Chapter Equations and Inequalities Section. Check Point Exercises. x + 9 x + 9 x x 6 Check: x + 9 (6) true The solution set is {6}.. (x+ ) 9 (x ) 8x+ 9 6x 8x 6x 8x 6 6x 6x x x + + x x Check: (x+ ) 9 (x ) [() + ] 9 [() ] [ + ] 9 [ ] [] 9 [] 9 true The solution set is {}... x x+ 7 x x ( x ) () ( x+ ) 7x x 7x x 7x+ x + x x Check: x x The solution set is {}. 7, x x 8 7 8x 8x x x 8x x 8 7x x x 7x 7 7 x The solution set is {}. 6 Copyright Pearson Education, Inc.

7 Section. Linear Equations and Rational Equations x., x x x x ( x ) ( x ) x x x ( x ) ( ) ( x ) x x 6 ( x ) 6 ( x ) 6 x+ x + x+ x x x The solution set is the empty set,. 6. Set y y. + x+ x x 6 + x+ x ( x+ )( x ) ( x+ )( x ) ( x+ )( x ) ( x+ )( x ) + x+ x ( x+ )( x ) ( x ) + ( x+ ) x + x+ x Check: + x+ x x true 7. x 7 ( x ) + x 7 ( x ) + x 7 x + x 7 x 7 The original equation is equivalent to the statement 7, which is false for every value of x. The solution set is the empty set,. The equation is an inconsistent equation. 8. x 7 ( x ) x+ 9 9( x+ ) x 7x+ 9 9x+ 9 x 7x+ 9 7x The original equation is equivalent to the statement 9 9, which is true for every value of x. The equation is an identity, and all real numbers are solutions. The solution set { xx is a real number}. D x+ 9 9 x x x + 9 x + 7 x 7 x.7 x.7 The formula indicates that if the low-humor group averages a level of depression of in response to a negative life event, the intensity of that event is.7. This is shown as the point whose corresponding value on the vertical axis is and whose value on the horizontal axis is.7. Concept and Vocabulary Check.. linear. equivalent. apply the distributive property. least common denominator;. 6. x 7. ( x+ )( x+ ) 8. x ; x 9. ( x+ ) + ( x+ ) x+ 9 Copyright Pearson Education, Inc. 7

8 Chapter Equations and Inequalities. identity. inconsistent Exercise Set.. 7x Check: 7x 7 7() The solution set is {}.. x (6x ) x 6x+ x + x 7 The solution set is {7}. Check: x (6x ) (7) [6(7) ] 77 ( ) 77 (7). x x x 7 6 The solution set is {}. Check: () x + x + 6 6x The solution set is {}. Check: 7() (x ) + 7 (x + ) x + + x + x + 9 The solution set is {9}. Check: (9 ) + 7 (9 + ) (7) + 7 () (x ) (x ) x + (x ) x + x + x + The solution set is { }. Check: ( ) ( ) + ( ) ( 9) ( 8) ( 7) (x ) (x 7) 6 x x + x 6 x The solution set is {6}. Check: 6 (6 ) (6 7) 6 () ( ) Copyright Pearson Education, Inc.

9 Section. Linear Equations and Rational Equations. [ y( y) ( y) [( y) y] [ yy6] 6y [yy] [y] 6y [y] y 6y y y98y7 6y y The solution set is { }. Check: [ + y ( y+ ) ( y ) [( y ) y+ ] [ + ( ) ( + ) [( ) ] [( ) ( ) + ] [ ()] [ ] [( ) ] [ 8] ( 9) [ + 9] ( 6) x x 7. x x 6 x x The solution set is {}. x x 9. x x 6 x + x x The solution set is {}.. x + x + 9x x+ 9x x The solution set is { }. Copyright Pearson Education, Inc. 9

10 Chapter Equations and Inequalities a. x x 6x x x x The solution set is {}. x+ x x+ x x+ 9+ 6x x 6x x The solution set is. x x + x x + + x The solution set is { }. x+ x+ 7 x+ x+ 7 7x x The solution set is. + ( x ) x x b.. a. b.. a. b. 7. a. b. 9. a. b. + x x 8 + 6x 6x x The solution set is. ( ) x + x + x + + x x 8+ x + The solution set is { }. + ( ) 6x x + 6x 8+ x 7x The solution set is {}. x x+ + (x ) x x x x+ + x x x + x + x The solution set is {}. ( ) x + x x + x x + ( x ) + x x x The solution set is {}. 6 Copyright Pearson Education, Inc.

11 Section. Linear Equations and Rational Equations. a. 8x 8 ( x ) x+ x+ 9. a. 6 ;( x,) x x+ ( x )( x+ ) b.. a. b.. a. 8x 8 x+ x+ 8x ( x+ ) 8 8x x+ 8 x no solution The solution set is the empty set,. + ( ) x x x + x x + ( x ) x + ( x ) + x The solution set is {}. 8 + ;( x,) x+ x ( x+ )( x ) b. 8 + x+ x ( x+ )( x ) ( x, x ) ( x ) + ( x+ ) 8 6+ x+ 8 x no solution The solution set is the empty set,. x 7. a. ( x, x ) x+ x x b. 6 x x+ x x 8 6 x x+ ( x )( x+ ) ( x, x ) ( x+ ) ( x ) 6 x+ x+ 6 x 6 no solution The solution set is the empty set,.. Set y y. (x 8) ( x ) + x x + x x x x + x 6 The solution set is {6}.. Set y y. x x x x ( x ) ( x ) x x+ x The solution set is { 7}. b. x x+ x x x x+ x ( x+ )( x ) ( x ) ( x+ ) x x x x The solution set is { }. Copyright Pearson Education, Inc. 6

12 Chapter Equations and Inequalities. Set y + y y. x x+ x+ x + 7x+ x x+ x+ ( x+ )( x+ ) x + 9 ( x+ )( x+ ) + ( x )( x ) x x ( x+ )( x+ ) ( x+ ) + ( x+ ) x+ 9 x+ + + x+ 9 8x+ 7 x+ 9 x 8 The solution set is {}. 7. [ x ( x)] 7( x+ ) [ x + x] 7x 7 [x ] 7x 7 8x 7x 7 x The solution set is {9}. 9. x + 6 x x + 6 ( x ) x x + 6 ( x ) ( x ) ( x ) x ( x )( x+ 6) ( x ) ( x ) ( x ) x ( x+ 6) ( x ) x+ 6 x+ 8 x The solution set is { }. 6 Copyright Pearson Education, Inc.

13 Section. Linear Equations and Rational Equations 6. x+ 9 9( x+ ) x x+ 9 9x+ 9 x x+ 9 x The solution set { xx is a real number}. The given equation is an identity. 6. ( x+ ) The solution set. The given equation is an inconsistent equation. 6. x+ 8x+ x + x The solution set {}. The given equation is a conditional equation. 67. x 6 + x x x 6+ ( x ) x 6+ x x 6 no solution The given equation is an inconsistent equation. 69. x+ x ( x+ ) (x ) + x 7 7 The solution set is { 7}. The given equation is a conditional equation. 7. x + x x ( x ) + x 6+ x x 8 no solution The solution set is the empty set,. The given equation is an inconsistent equation. 7. 8x ( + ) + 8x + 8 The solution set is { }. The given equation is a conditional equation x 8+ x 8 8 The solution set is {8}. The given equation is a conditional equation. 7 + x x+ ( x+ )( x ) ( x+ ) + ( x ) 7 x x 7 The solution set is { }. The given equation is a conditional equation. x x + 6 ; x, x+ x x 9 x( x ) ( x+ ) x + 6 x x x 6 x + 6 x x 6 x + 6 x 6 6 x 7 No solution The solution set is { }. The given equation is an inconsistent equation. 8. The equation is ( x ) ( x), and the solution is. 8. The equation is ( x ) ( x), and the solution is.. Copyright Pearson Education, Inc. 6

14 Chapter Equations and Inequalities 8. Solve: ( x ) + x ( x) x 8+ x + x x 6 6x x 6 x Now, evaluate x x for : x ( ) ( ) ( ) Solve for x: ( x + ) x + 6 ( x+ ) (x+ 6) + 9 x+ 7x + 9 7x Solve for y: y y+ 8 7y 8 7y 8 y Now, evaluate x xy y x ( xy y) ( ) [ ( ) ( ) ] ( ) [ ( ) ] 9 (+ ) ( ) + 6 x ( 9 ) x ( 8 ) x 7 x 8 x x The solution set is { }. ( ) for and y : 9. ( ) 8 7x 6 x x 6 ( + ) + x 8 7x [ x] 8 7x [ 7+ x] 8 7x [ + x] 8 7x x x x 6 6 The final statement is a contradiction, so the equation has no solution. The solution set is. 9..7x+.().( x+ ).7x+ 8.x+.x + 8.x The solution set is {}. { } x+ x x ( x 6) 9. ( ) { [ ]} { [ ]} { } { } x+ x x x x+ x x 7 x x+ x x+ 7 x x+ x+ 7 x x+ + x 7 x 6x x x x 8 The solution set is { }. x 97. a. p + 7 p + 7 p + 7 p 7 According to the model, 7% of American adults smoked cigarettes in. This underestimates the value shown in the bar graph by %. 6 Copyright Pearson Education, Inc.

15 Section. Linear Equations and Rational Equations x b. p + 7 x 7+ 7 x 6 According to the model, only 7% of American adults will smoke cigarettes 6 years after 97, or. 99. a. What cost $, in 967 would cost about $, in. b. C 88x+,96 88() +, 96,7 According to Model, what cost $, in 967 would cost about $,7 in. This describes the estimate from part (a) reasonably well. x +.(). C x + x +.().8 x +.8( x+ ) x+.().8x+ x+.7x 9.7x liters of pure peroxide must be added. 7.. Answers will vary. 7. { } c. C + 8x+,68 () + 8() +,68, 68 According to Model, what cost $, in 967 would cost about $,68 in. This describes the estimate from part (a) reasonably well. 9. { }. C 88x+,96 77,77 88x +,96,7 88x 8 x Model predicts the cost will be $77,77 8 years after 98, or 8.. learning trials; represented by the point (,.9 ) on the graph.. makes sense. makes sense. false; Changes to make the statement true will vary. A sample change is: In the first equation, x. 7. false; Changes to make the statement true will vary. A sample change is: If a, then ax + b is equivalent to b, which either has no solution (b ) or infinitely many solutions (b ). Copyright Pearson Education, Inc. 6

16 Chapter Equations and Inequalities 9. 7x + + x b 7( 6) b b b 8 9 b 8 9b b. Let the number of text messages at which the costs of the two plans are the same. Plan A Plan B +.8x +.x +.8x +.x.8x.x.8x..x.x.x.x.. The two plans are the same at text messages.. x +. +.x. x + Section. Check Point Exercises. Let the median starting salary, in thousands of dollars, of education majors. Let x + the median starting salary, in thousands of dollars, of computer science majors. Let x + the median starting salary, in thousands of dollars, of economics majors. x+ ( x+ ) + ( x+ ) x+ x+ + x+ + x + 6 x + 9 The median starting salary of education majors is $ thousand, of computer science majors is $6 thousand, and of economics majors is $9 thousand.. Let the number of years after x.9x 8.9x x 67 % of freshmen will respond this way 67 years after 969, or 6.. Let the computer s price before the reduction. x. 8.7x Before the reduction the computer s price was $.. Let the amount invested at 9%. Let the amount invested at %..9x+.( x) 87.9x+.x 87.x + 87.x $ was invested at 9% and $8 was invested at %. 6. Let the width of the court. Let x + the length of the court. l+ w P ( x+ ) + x 88 x+ 88+ x 88 x x x + 9 The dimensions of the court are feet by 9 feet. 66 Copyright Pearson Education, Inc.

17 Section. Models and Applications 7. l+ w P l+ w l P l w P l w P l P l w 8. P C+ MC P C( + M) P C( + M) + M + M P C + M P C + M Concept and Vocabulary Check.. x x. +.x. x.x or.8x..x+.9(, x) 6. isolated on one side 7. factoring Exercise Set.. Let the number x 6 x 6 The number is 6.. Let the number x.x.8x The number is.. Let the number.6x+ 9.6x 9 The number is. 7. Let the number.7x The number is. 9. Let the number x + 6 the other number x+ ( x+ 6) 6 x+ x+ 6 6 x x 8 9 If 9, then x + 6. The numbers are 9 and.. y y ( ) (x+ ) x 8x 8x 6 8x y 8y + (x ) 8(x+ ) + x 6x+ 8 + x 6x+ x x 8 y y y. (x+ 6) ( x+ 8) ( x) 6x+ 8 x x x x x + The solution set is the set of all real numbers. Copyright Pearson Education, Inc. 67

18 Chapter Equations and Inequalities y + y y 7. x + x x + x x x x x xx ( ) xx ( ) x x ( x ) x x x 9. Let the number of years spent watching TV. Let x + 9 the number of years spent sleeping. x+ ( x+ 9) 7 x+ x+ 9 7 x x 8 9 x Americans will spend 9 years watching TV and 8 years sleeping.. Let the average salary, in thousands, for an American whose final degree is a bachelor s. Let x 9 the average salary, in thousands, for an American whose final degree is a master s. x+ (x 9) 6 x+ x x 9 6 The average salary for an American whose final degree is a bachelor s is $ thousand and for an American whose final degree is a master s is $6 thousand.. a. y, b. y, 9, 9,,, The car s value will drop to $9 after years. c. Graph: 7. Let the number of months. The cost for Club A: x + The cost for Club B: + x+ + x + x The total cost for the clubs will be the same at months. The cost will be () + () + $6 9. Let the number of uses. Cost without discount pass:.x Cost with discount pass: +.7x.x +.7x.x The bus must be used times in a month for the costs to be equal.. Let the number of years since..x 67.x 67.x 6 6 x. x 67% of American adults will view college education as essential years after, or. 68 Copyright Pearson Education, Inc.

19 Section. Models and Applications. a. Let the number of years (after ). College A s enrollment:, + x College B s enrollment: 6,8 x, + x 6,8 x, + x 6,8 x, 9 The two colleges will have the same enrollment 9 years after, or 9. That year the enrollment will be, + (9) 6,8 (9), students b. Check points to determine that y, + x and y 6,8 x.. Let the cost of the television set. x.x 6.8x 6 The television set s price is $.. Let the nightly cost x+.8x 6.8x 6 The nightly cost is $. 7. Let c the dealer s cost 8 c+.c 8.c 67. c The dealer s cost is $ Let the amount invested at 6%. Let 7 the amount invested at 8%..6x+.8(7 x).6x+ 6.8x.x + 6.x. 7 $ was invested at 6% and $ was invested at 8%.. Let amount invested at % 8 amount invested at % loss. x.(8 x) 6.x +.x 6.7x 6 8-x $6 at %, $ at % loss. Let w the width of the field Let w the length of the field P length + width ( ) ( ) ( w) ( w) + w+ w 6w w If w, then w. Thus, the dimensions are yards by yards.. Let w the width of the field Let w + 6 the length of the field 8 6w + 6 6w 6 w If w 6, then w + 6 (6) Thus, the dimensions are 6 feet by 78 feet. 7. Let the width of the frame. Total length: 6 + x Total width: + x P (length) + (width) 7 6 ( + x) + ( + x) 7 + x+ + x 7 8x x x The width of the frame is inches. 9. Let number of hours labor cost x It took hours. Copyright Pearson Education, Inc. 69

20 Chapter Equations and Inequalities. Let inches over feet + 7 A height of feet 7 inches corresponds to pounds.. Let the weight of unpeeled bananas. 7 weight of peeled bananas x The banana with peel weighs 7 ounces.. A lw A w l area of rectangle 7. A bh A bh A b ; h area of triangle 9. I Prt I P ; rt interest 6. E mc E m ; c Einstein s equation 6. T D+ pm T D pm T D pm m m T D p m total of payment 6. A ha ( + b ) A ha ( + b) A a + b h A b a h area of trapezoid 67. S P + Prt S P Prt S P r; Pt interest F B S V BS ( V) F F S V B F S + V B IR + Ir E I( R+ r) E E I R + r electric current + p q f qf + pf pq f( q+ p) pq pq f p+ q thin lens equation Answers will vary. 8. does not make sense; Explanations will vary. Sample explanation: Though mathematical models can often provide excellent estimates about future attitudes, they cannot guaranty perfect precision. 8. does not make sense; Explanations will vary. Sample explanation: Solving a formula for one of its variables does not produce a numerical value for the variable. 7 Copyright Pearson Education, Inc.

21 Section. Complex Numbers 8.. x+.9( x).+ 9.9x.8x students at the north campus, students at south campus. 87. Let woman s age Coburn s age + (x + ) + x + x + Coburn is 6 years old the woman is years old. 89. Let mother s amount boy s amount girl s amount x x+ x+, 7, $, The mother received $, the boy received $8, and the girl received $ C S V C N L VL CL CN + SN VL SN CL CN VL SN C( L N) VL SN C L N VL SN C L N (7 x)( x) x+ 6x+ x 9x+ x or x 9x Section. Check Point Exercises. a. ( i) + ( + i) i+ + i ( + ) + ( + ) i 8 + i. a.. b. ( + 6 i) ( i) + 6i + i ( ) + (6 + ) i + 7i b. 7( i 9) i 7() i 7(9) i i i 6i i 6( ) 6+ i ( + i)(6 7 i) i+ i 8i i+ i 8( ) + 8 i+ i 8 i + i + i + i i i + i + i+ 6i+ i 6 + i i i + i i i 7 7 Copyright Pearson Education, Inc. 7

22 Chapter Equations and Inequalities. a i 7 + i 8 i 9 + i 6 i + i 7i. 6 ( + i) ( i) 6+ i+ + i i 7. 8i ( 9i) 8i + 9i + 8i + 9i + 7i b. c. ( + ) ( + i ) ( ) + ( )( i ) + ( i ) i + i i + ( ) i + + i + i i + 7+ i 9. i(7i ) i + i ( ) + i + i. ( + i)( + i) i+ i+ i + 7i 9 + 7i. (7 i)( i) i+ i+ i i 9 i. ( + i)( i) 9 i+ i i 9+ Concept and Vocabulary Check.. ; 7. ( + i)( i) + i i i + 6. complex; imaginary; real. 6i. i. 8 ; i ; i ; i 7. + i 8. i; i Exercise Set. i ;. (7 + i) + ( i) 7 + i + i i i 8 i + i + i+ 9i + i 9 + i 9. ( ).. + i i i + i ( + i) 9+ ( + i) + i + i i i i i i + i + i + i + i i +. ( + i) ( 7i) + i + 7i + i + 7i + 9i 7 Copyright Pearson Education, Inc.

23 Section. Complex Numbers. 7. 8i 8i + i i i + i i+ i i + i + i + i i + i + i i + i i + 7+ i 7 + i 9. 6 i 6 i 8i i i (i) + (9i) i + 7i 7i. ( ) + ( + i) 8i+ i 8i 8i. ( 7) ( i 7) 7. + i + i i 7 + 6i 7 ( ) i 8+ i 6 8+ i + i i i 8 6 i 8 i 8 8 i 8( i ). ( ) ( i ) i 6 i. ( )( ) ( i )( 8i ) i. ( i)( i) ( i)( + i) ( i i i ) ( i ) + i+ i 9+ i 7 i+ i 7 i + ( ) i 7. ( + i) ( i) ( i i ) ( 9 6i i ) i+ i 9+ 6i i + i i+ 9 i i+ 7i 7 i or + 7i f x x+ f ( + i) ( + i) ( + i) + + i+ i i+ + i. ( ) Copyright Pearson Education, Inc. 7

24 Chapter Equations and Inequalities. f ( x) f ( i) x + 9 x ( i ) + 9 i 9i + 9 i 9+ 9 i i + i i + i + i 9i + i i + i. E IR ( i)( + 7i) E + 8i i i E + i ( ) E + + i 7 + i The voltage of the circuit is (7 + i) volts. 7. Sum: ( + i ) + ( i ) + i + i + Product: ( + i )( i ) i + i i makes sense 69. does not make sense; Explanations will vary. Sample explanation: i ; It is not a variable in this context. 7. false; Changes to make the statement true will vary. A sample change is: All irrational numbers are complex numbers. 7. false; Changes to make the statement true will vary. A sample change is: 7+ i 7+ i i 6i i + i + i i ( + i)( i) 6 i+ i i 6 + i i 7 i 7+ i 7 i 8 i 9 i 8 i i 8 i i Answers will vary. 7 Copyright Pearson Education, Inc.

25 Section. Quadratic Equations i + + i i i 8 + i i 8i + i 8i i + i i 6i 8i i 6i i i x + 7x (x )( x+ ) x 6x+ 9 ( x )( x ) ( x ) b b ac (9) (9) ()( ) a () Section. Check Point Exercises. a. 9x ( xx ) or x The solution set is {, }. b.. a. b. c. x + x + x (x )( x+ ) x or x+ x The solution set is,. 7 ± 7 The solution set is { 7, 7}. x + x 9 ± 9 ± i The solution set is {, i i} ( ) x + x + ± ±. The solution set is { +, }.. a. The coefficient of the x-term is 6. Half of 6 is, and is 9. 9 should be added to the binomial. x + 6x+ 9 ( x+ ) b. The coefficient of the x-term is. Half of is, and is. should be added to the binomial. x x+ x Copyright Pearson Education, Inc. 7

26 Chapter Equations and Inequalities.. c. The coefficient of the x-term is. Half of is, and is 9. should be added to the binomial. 9 x + x+ x+ 9 x + x x + x x + x+ + x + ( ) x + ± ± The solution set is { ± }. x + x + x x x + x x + 6 x + ± 6 x + ± ± ± ± The solution set is x + x a, b, c b b ac ± a ± () ± + 8 ()( ) ± ± ( ± ) ± ± The solution set is. x x+ a, b, c b± b ac a ( ) ± ( ) ()() () ± 8 ± ± i ± i The solution set is { + i, i}. 8. a. a, b 6, c 9 b ac (6) ()(9) 6 6 Since b ac, the equation has one real solution that is rational. 76 Copyright Pearson Education, Inc.

27 Section. Quadratic Equations 9. b. a, b 7, c b ac ( 7) ()( ) Since b ac >, the equation has two real solutions. Since 8 is a perfect square, the two solutions are rational. c. a, b, c b ac ( ) ()() 8 Since b ac <, the equation has two imaginary solutions that are complex conjugates. P.A +.A+ 7.A +.A+ 7.A +.A 8 a., b., c 8 b± b ac A a ± A (.). ±. A. (.) (.) (.)( 8) A A.. A 6 A Age cannot be negative, reject the negative answer. Thus, a woman whose normal systolic blood pressure is mm Hg is approximately 6 years old. Concept and Vocabulary Check.. quadratic. A or B. x-intercepts. ± d. ± b± b ac a. ; 9;. ; ;. ±.. ± i b 6 ac. Let w the screen s width. w + l d w + w + 6 w w ± w ± Reject the negative value. The width of the television is inches. 6. no 7. two 8. the square root property 9. the quadratic formula. factoring and the zero-product principle. right; hypotenuse; legs. right; legs; the square of the length of the hypotenuse Copyright Pearson Education, Inc. 77

28 Chapter Equations and Inequalities Exercise Set x (x + )(x ) x+ or x or The solution set is {, }. 8x x 8x+ (x )(x ) x or x or The solution set is {, }. 6x + x (x + )( ) x+ or x or The solution set is,. x 8 x 8 ( + )(x ) + or x or The solution set is,. + x ( xx+ ) or x+ or The solution set is {, }. 7 x x x x. ( ) x 6x x + 7x + x( + ) or + The solution set is, ( + )(x ) 7 7 x 7 7x + x+ 6x+ 9 (x + )(x ) x+ or x or The solution set is {, } ± 9 ± The solution set is {, }. x + ± The solution set is {, }. x x ± ± i The solution set is { i i}. ( x + ),. x + ± x + ± ± + or 7 The solution set is { 7, }. 78 Copyright Pearson Education, Inc.

29 Section. Quadratic Equations. ( x ) ( x ) x ± ± The solution set is { +, }.. ( x + ) 6 x + ± 6 x+ ± i ± i The solution set is { + i i} 7. ( x ) x ± x ± i ± i,. The solution set is { + i i } 9. ( + ) 9,. + ± 9 ± + or + or The solution set is,. ( x ) 7 x ± 7 ± 7 ± The solution set is,.. ( ) 8 ± 8 ± ± ± + The solution set is, x + x 6 6 x + x+ 6 ( x+ 6) x x x x+ ( x ) x x x x x 7x x x x x x x x x x x x x x x + 6x 7 x + 6x ( x + ) 6 x + ± ± The solution set is { 7, }. Copyright Pearson Education, Inc. 79

30 Chapter Equations and Inequalities 9. x x x x+ + ( x ) x ± ± The solution set is { +, }.. x 6x x 6x x 6x ( x ) x ± ± The solution set is { +, }.. x + x+ x + x x + x+ + ( x + ) x + ± ±. The solution set is { +, }. x x+ 6 x x 6 x x+ 6+ x x ± x ± ± + or The solution set is {, }. 7. x + x x x + x + ± ± + The solution set is,. 9. x 7x+ 7 x x+ 7 x x x x 6 7 x ± 7 ± The solution set is,. 8 Copyright Pearson Education, Inc.

31 Section. Quadratic Equations x x x x x x x x x+ + x ± x ± + The solution set is,. x x x x x x x 9 ± 7 x ± The solution set is,. x + 8x+ 8± 8 ()() () 8± 6 6 8± 8± The solution set is {, } x + x+ ± ()() () ± ± + The solution set is,. ± ( ) ()( ) () ± ± The solution set is,. 6 6 x x+ 7 x x 7 ± ( ) ()( 7) () ± + 8 ± 6 8 ± 9 8 ± The solution set is,. Copyright Pearson Education, Inc. 8

32 Chapter Equations and Inequalities x 6x+ 6 ± ( 6) ()() () 6± 6 6± 6± i ± i The solution set is { + i i} x x ( ) ()( ), ; unequal real solutions x x+ ( ) ()() 97; unequal real solutions x x+ ( ) ()() ; real solution x 7 ( ) ()( 7) ; unequal real solutions x x x (x+ )( x ) x+ or x x or The solution set is,. 8. x + x x x+ (x )( x ) x or x x or The solution set is, ± ± The solution set is {, }. x x x x+ + ( x ) x ± ± The solution set is { +, }. (x+ )( x+ ) x + 8x+ + x + x+ ± () ± 88 ± ()() + The solution set is,. 8 Copyright Pearson Education, Inc.

33 Section. Quadratic Equations 9. ( ) 6 ± 6 ± ± 8 or 8 or 8 The solution set is, x x+ x x+ ( x )( x ) x The solution set is {}. x 6 x 6 x ± The solution set is { } x 6x+ x 6x x 6x ( ) x x ± i,. ± i + i, i. The solution set is { }. x 7 x x 7 x x+ 7+ ( ) x x ± i ± i The solution set is { + i i },. x 7x. x(x 7) or x 7 x 7 7 or 7 The solution set is,.. + ; x, x x x + x x x 6 ± ( ) ( ) ( )( 6) () ± 6+ ± ± ± The solution set is { +, }. 7. x ; x, x x+ x 9 x( x+ ) + 6( x ) 8 x + 6x+ 6x 8 8 x + x+ x + 6x+ ( x+ )( x+ ) The solution set is {, }. 9. x x ( x+ )( x ) x+ x or This equation matches graph (d). Copyright Pearson Education, Inc. 8

34 Chapter Equations and Inequalities. ( x + ) + ( x + ) x + ± ±, This equation matches graph (f).. x x+ a, b, c b± b ac a ± () ( ) ( ) ()() ± ± i ± i This equation has no real roots. Thus, its equation has no x-intercepts. This equation matches graph (b).. y x x x x (x+ )( x ), 7. yy ( x )( x+ ) x + x + 8 ( x+ 6)( x ) 6, 8 Copyright Pearson Education, Inc.

35 Section. Quadratic Equations 9. y + y x + x+ x+ x x+ x+ x( x+ )( x+ ) ( x+ )( x+ ) + x+ x+ + + x x x x x ( x+ )( x+ ) + ( x+ )( x+ ) ( x )( x ) ( + ) + ( + ) ( + )( + ) x + x+ x+ x + x+ x + x b± b ac a ± () ± () () ()( ) + The solution set is,.. y y x + x x + x ( ) ( ) x + x + x x+ b b ac x+ 6 ± a ( ) ± ( ) ()(6) () ± 8 6 ± 7 6 ± The solution set is,. Copyright Pearson Education, Inc. 8

36 Chapter Equations and Inequalities. Values that make the denominator zero must be excluded. x + x 9 b± b ac a () ± () ()( 9) () ± 88 ± ±. x ( 6+ x) x x 6 Apply the quadratic formula. a b c 6 ( ) ( ) ( )( 6) () ± ( ) ± ± 8 ± 7 ± 7 ± 7 We disregard 7 because it is negative, and we are looking for a positive number. Thus, the number is x + x + x + ( x )( x ) x+ ( x+ )( x ) Multiply both sides of the equation by the least common denominator, ( x )( x )( x+ ). This results in the following: Apply the quadratic formula: a b c. ± ()( ) ± () ± ( ) The solutions are ±. ± 9. x + Apply the quadratic formula: a b c ( )( ) ( ) ±, and the solution set is ± 9 ( 6) ± ± Evaluate the expression to obtain two solutions. + or 8 8 The solutions are and set is,., and the solution x+ ( x )( x ) + ( x ) x+ x x x+ + x x+ x + x x + x 86 Copyright Pearson Education, Inc.

37 Section. Quadratic Equations. x x N x x x x x x ( x+ 6)( x 7) x+ 6 or x Reject the negative value. There were 7 players.. This is represented on the graph as point (7, ). f..9x+ 8.. ( )..9x x+. Apply the quadratic formula. a. b.9 c. x (.9) (.9) (.)(.). ( ) ±.9 ± ± ± or.9 The solutions are approximately.9 and 8.. Thus, year olds and 8 year olds are expected to be in fatal crashes per million miles driven. The function models the actual data well. 9. a.. b. Φ Φ Φ Φ ( Φ ) ( Φ ) Φ Φ Φ Φ Φ b± b ac Φ a ( ) ± ( ) ()( ) Φ () ± + Φ ± Φ, reject negative + Φ c. The golden ratio is ± ± to. We disregard because we can t have a negative measurement. The path is miles, or approximately. miles. 7. Let y x x x x 6 x x ± 7.8 The distance is 7.8 feet. Using the TRACE feature, we find that the height of the shot put is approximately feet when the distance is 77.8 feet. Graph (b) shows the shot path. Copyright Pearson Education, Inc. 87

38 Chapter Equations and Inequalities. Let w the width Let w + the length Area lw ( w+ ) w w + w w + w ( w+ 9)( w 6) Apply the zero product principle. w+ 9 w 6 w 9 w 6 The solution set is { 9,6 }. Disregard 9 because we can t have a negative length measurement. The width is 6 feet and the length is 6+ 9feet. 7. Let the length of the side of the original square Let x + the length of the side of the new, larger square ( x + ) 6 x + 6x+ 9 6 x + 6x ( x+ )( x ) Apply the zero product principle. x+ x The solution set is {, }. Disregard because we can t have a negative length measurement. This means that x, the length of the side of the original square, is inches. 9. Let the width of the path ( + x)( + x) 6 + x+ x+ x 6 + 6x+ x 6 x + 6x+ 6 x + 6x ( x + x ) x+ x ( )( ) Apply the zero product principle. ( x + ) x x+ The solution set is {, }. Disregard because we can t have a negative width measurement. The width of the path is meters.. xx ( )() x ± The length and width are inches.. x( x) x x x x+ ± ± 96 ± 7. 9.,.7 9. in and.7 in. 6. Answers will vary. ( ) ( ) ( )( ) ( ) 67. does not make sense; Explanations will vary. Sample explanation: Higher degree polynomial equations can have only one x-intercept. 69. makes sense 7. true 7. false; Changes to make the statement true will vary. A sample change is: The first step is to collect all the terms on one side and have on the other t + v t s a 6, b v, c s s t + vt 6 ( ) ( 6)( ) ( 6) ( ) 6 v ± v s t v ± v s t v ± v 6s t 88 Copyright Pearson Education, Inc.

39 Mid-Chapter Check Point x + x x x ( x+ ) ( x+ ) ( x+ )( x ) ( x+ )( x+ )( x ) ( x+ + ) x+ + ( x+ )() + x+ + x+ + x+ + x+ x + x + ( 8) + ( 8) + ( ) + ( ) + () + ( ) + +, true The statement is true. Mid-Chapter Check Point. + ( x+ ) ( ) + + 6x 8 + 6x The solution set is { } x x 7 x x 7 (x 7)( x+ ) x 7 or x+ x The solution set is,... x x x x ( x ) ( x ) () ( x ) ( x ) x x x x 7 7 The solution set is { 7. } 6x b± b ac a ± () 6± 6 6 ( 6) ( 6) ()( ) 6± 6 ± + The solution set is,.. x ( x) (x+ ) x ( x) (x+ ) x + 6x+ 6x 6x The equation is an identity. The solution set is { xx is a real number }. Copyright Pearson Education, Inc. 89

40 Chapter Equations and Inequalities 6. x + 7 x 6 x ± 6 ± 6 ± 6 ± 6 6 The solution set is,. 7. x(x ) x x + b± b ac a ± () ± ( ) ( ) ()() ± i + i i The solution set is, x x + x x + x x ( x) 6x 6( x) 6() + 6() x x+ 6 8x 9 x+ 6 8x The solution set is { } ( x + ) x + ± ± 6 The solution set is { + 6, 6 }.. x x + x + x () x x x x + x x x+ x x+ b± b ac a ± () ± ± ± ( ) ( ) ()() The solution set is { +, }. 9 Copyright Pearson Education, Inc.

41 Mid-Chapter Check Point. + ( x ) x x+ 6 x 6 The solution set is. x x., x, x x + 6x+ 8 x+ x+ x x ( x+ )( x+ ) x+ x+ xx ( + )( x+ ) x ( x+ )( x+ ) ( x+ )( x+ ) x+ x+ xx ( + )( x+ ) ( x+ )( x+ ) x x+ x+ x( x+ ) ( x+ ) x + x x 8 x x 8 ( x+ )( x ) x+ or x must be rejected. The solution set is {. }. Let y. x + 6x+ b b ac ± a ± () 6± 8 x x (6) (6) ()() 6± 7 ± 7 x-intercepts: + 7 and 7.. Let y. ( x+ ) x (6 x) x+ 6+ x x x x-intercept:.. Let y. x + 6 x 6 ± ± i There are no x-intercepts. 6. Let y. x x + x x 6() 6 + x x + x x b± b ac a ± () ± () () ()( ) x-intercepts: 7. Let y. x x+ 8 b± b ac a ± () ± 7 ± i 7 There are no x-intercepts. + and. ( ) ( ) ()(8) 8. y y (x ) (x+ ) ( x+ ) 6x 8x x x 7 x 7 The solution set is {. } Copyright Pearson Education, Inc. 9

42 Chapter Equations and Inequalities 9. yy (x+ )( x+ ) x + 7x+ 6 x + 7x (x )( x+ ) x or x + The solution set is,.. x + x x + Since b, we add x + x+ + ( x + ) 8 Apply the square root principle: x + ± 8 x + ± 7 ± 7 ± 7 The solutions are ± 7 { ± 7}.., and the solution set is. x + x+ a b c b ac ( )( ) 7 Since the discriminant is negative, there are no real solutions. There are two imaginary solutions that are complex conjugates.. x( x ) + x x + x x x+ a b c ( ) ( )( ) b ac 6 6 Since the discriminant is positive and a perfect square, there are two rational solutions.. x ( x, y). x ( x, y). x ( x, y) Copyright Pearson Education, Inc.

43 Mid-Chapter Check Point 6. L a+ ( n ) d L a+ dn d dn a d L dn a d L d d d d a L n + + d d L a n + d d L a n + d 7. A lw + lh + wh lw lh wh A l( w h) wh A wh A l w h A wh l w+ h 8. ff f f + f ff f+ f ff + ff ff ff ff ff f( f f) f f f f f f f ff ff f or f f f f f ( f + f )( f ) ( f + f ) 9. Let the number of times sorry was used. Let x + 9 the number of times love was used. Let x + the number of times thanks was used. x+ ( x+ 9) + ( x+ ) 8 x+ x+ 9 + x x x + The word sorry was used times, the word love was used 6 times, and the word thanks was used times.. Let the number of years since 96..8x.8x.8x.8.8 x 8 If this trend continues, corporations will pay zero taxes 8 years after 96, or.. Let the amount invested at 8%. Let, the amount invested at 9%..8x+.9(, x).8x+.9x.x +.x.,,, $, was invested at 8% and $, was invested at 9%.. Let the number of text messages. Plan A: C +.x Plan B: C +.7x Set the costs equal to each other. +.x +.7x +.x.x x The cost will be the same for text messages. C +.x C +.() The cost for text messages will be $.. Let the price before the reduction. x.x 68.6x 68.6x The price before the reduction was $78. Copyright Pearson Education, Inc. 9

44 Chapter Equations and Inequalities. Let the amount invested at %. Let the amount invested that lost %..x.( x).x +..7x.7x $ was invested at % and $ lost %.. Let the width of the rectangle Let x + the length of the rectangle l+ w P (x+ ) + x 6 x+ + x 6 6x + 6 6x 6 6x x + 7 The dimensions of the rectangle are 6 ft by 7 ft. 6. Let the width of the rectangle Let x the length of the rectangle lw A (x ) 8 x 8 x x 8 (x+ 7)( x ) x+ 7 or x x must be rejected. If, then x 7 The dimensions of the rectangle are ft by 7 ft. 7. Let the height up the pole at which the wires are attached. x + x + 69 ± must be rejected. The wires are attached yards up the pole. 8. a. 9. P x + 7x+ 99 x + 7x+ x + 7x 9 x 9x+ 98 ( x 6)(x 8) x 6 or x 8 6 x 8. The population reached 99 after 6 years. b. This is represented by the point (6, 99). P.x.7x+..x.7x+..x.7x.9 b± b ac a (.7) ± (.7) (.)(.9) (.).7 ± x 6, x (rejected) The percentage of foreign born Americans will be % about 6 years after 9, or 6.. (6 i) (7 i) 6 i 7 + i i... ( + ) i i i i i ( + i)( i) i+ i i + i+ 7+ i + i + i + i + i+ i+ i i i + i i + i + i i. 7 i i i. ( ) ( i ) i + i i i 9 Copyright Pearson Education, Inc.